a bag contains 8 red marbles 7 black and 5 white marbles. of 3 marbles are drawn at random, find the probability that
To find the probability of drawing 3 marbles of a specific color, we need to determine the total number of possible outcomes and the number of favorable outcomes.
Total number of possible outcomes:
There are a total of 20 marbles in the bag (8 red + 7 black + 5 white). When drawing 3 marbles without replacement, the total number of possible outcomes is given by the combination formula, which is written as nCr. In this case, we have 20 marbles, and we are drawing 3, so the total number of possible outcomes is:
nCr = (n!) / (r!(n-r)!)
20C3 = (20!) / (3!(20-3)!)
= (20!) / (3!17!)
= (20 * 19 * 18 * 17!) / (3 * 2 * 1 * 17!)
= (20 * 19 * 18) / (3 * 2 * 1)
= 1140
So, there are 1140 total possible outcomes.
Number of favorable outcomes:
We are interested in drawing 3 marbles of a specific color. Let's say we want to find the probability of drawing 3 red marbles. There are 8 red marbles in the bag, so the number of ways to choose 3 red marbles is:
8C3 = (8!) / (3!(8-3)!)
= (8!) / (3!5!)
= (8 * 7 * 6) / (3 * 2 * 1)
= 56
So, there are 56 favorable outcomes for drawing 3 red marbles.
Probability:
The probability of an event occurring is given by the ratio of the number of favorable outcomes to the number of total possible outcomes. Therefore, the probability of drawing 3 red marbles can be calculated as:
P(3 red marbles) = Number of favorable outcomes / Total number of possible outcomes
= 56 / 1140
= 14 / 285
So, the probability of drawing 3 red marbles is 14/285 or approximately 0.0491.
To find the probability of drawing certain marbles, we need to know the total number of outcomes and the favorable outcomes. In this case, the total number of outcomes is the number of ways to draw 3 marbles from the bag.
The total number of marbles in the bag is 8 red + 7 black + 5 white = 20 marbles.
So, the total number of ways to draw 3 marbles from 20 is given by the combination formula, C(n, r), which is calculated as C(20, 3) = 20! / (3! * (20 - 3)!), where n is the total number of marbles and r is the number of marbles being drawn.
Now let's determine the favorable outcomes, which is the number of ways to draw 3 marbles such that they are all red.
Since there are 8 red marbles in the bag, the number of ways to draw 3 red marbles is given by the combination formula, C(8, 3) = 8! / (3! * (8 - 3)!).
Finally, we can calculate the probability by dividing the number of favorable outcomes by the total number of outcomes:
Probability = (Number of favorable outcomes) / (Total number of outcomes)
= C(8, 3) / C(20, 3)
Now, let's calculate the probabilities:
Number of favorable outcomes = C(8, 3) = 8! / (3! * (8 - 3)!) = (8 * 7 * 6) / (3 * 2 * 1) = 56
Total number of outcomes = C(20, 3) = 20! / (3! * (20 - 3)!) = (20 * 19 * 18) / (3 * 2 * 1) = 1140
Probability = 56 / 1140 ≈ 0.0491
So, the probability of drawing 3 marbles at random and all of them being red is approximately 0.0491 or 4.91%.